Math used in computer algebra systems

I am recently very interested in computer algebra systems and was thinking about the maths in it.
Basic things that CAS does is algebra(solve set of equations), integration, limits, series etc.

Various algorithms are used for such operations. For e.g.:- for integration there is an algorithm called risch algorithm.

Now, these algorithms seemed very new to me. They are different from methods we use in solving such problems. Why do we need a separate approach in solving math for computers? Why can't we use normal math algorithms/methods?

For one reason, CAS evolved to manipulate symbols, rather than evaluate expressions numerically. It's a different approach because the goal is different. CAS can be extended to do limited numerical evaluations, since numbers can be represented symbolically.